Reinhard Lerch

Simulation of piezoelectric devices by two- and three-dimensional finite elements

A method for the analysis of piezoelectric media based on finite-element calculations is presented in which the fundamental electroelastic equations governing piezoelectric media are solved numerically. The results obtained by this finite-element calculation scheme agree with theoretical and experimental data given in the literature. The method is applied to the vibrational analysis of piezoelectric sensors and actuators with arbitrary structure. Natural frequencies with related eigenmodes of those devices as well as their responses to various time-dependent mechanical or electrical excitations are computed. The theoretically calculated mode shapes of piezoelectric transducers and their electrical impedances agree quantitatively with interferometric and electric measurements. The simulations are used to optimize piezoelectric devices such as ultrasonic transducers for medical imaging. The method also provides deeper insight into the physical mechanisms of acoustic wave propagation in piezoelectric media.<<ETX>>

Finite element analysis of piezoelectric transducers

A finite element technique for modeling the vibrational behavior of arbitrarily shaped piezoelectric transducers immersed in an acoustic fluid is presented. The elastic and electrical responses of the piezoelectric structure are computed by piezoelectric finite elements, and the wave propagation in the ambient acoustic medium is computed by acoustic finite elements. The acoustic feedback of the surrounding acoustic fluid to the piezoelectric solid is considered. This method makes it possible to analyze piezoelectric devices with respect to their mechanical strains and stresses, electrical fields and displacements, and various integral properties, such as the electric input impedance and the electromechanical coupling coefficient. The application of this method to ultrasonic transducers, especially those used in array antennas, is reported.<<ETX>>

Finite-element simulation of wave propagation in periodic piezoelectric SAW structures

Many surface acoustic wave (SAW) devices consist of quasiperiodic structures that are designed by successive repetition of a base cell. The precise numerical simulation of such devices, including all physical effects, is currently beyond the capacity of high-end computation. Therefore, we have to restrict the numerical analysis to the periodic substructure. By using the finite-element method (FEM), this can be done by introducing periodic boundary conditions (PBCs) at special artificial boundaries. To be able to describe the complete dispersion behavior of waves, including damping effects, the PBC has to be able to model each mode that can be excited within the periodic structure. Therefore, the condition used for the PBCs must hold for each phase and amplitude difference existing at periodic boundaries. Based on the Floquet theorem, our two newly developed PBC algorithms allow the calculation of both, the phase and the amplitude coefficients of the wave. In the first part of this paper we describe the basic theory of the PBCs. Based on the FEM, we develop two different methods that deliver the same results but have totally different numerical properties and, therefore, allow the use of problem-adapted solvers. Further on, we show how to compute the charge distribution of periodic SAW structures with the aid of the new PBCs. In the second part, we compare the measured and simulated dispersion behavior of waves propagating on periodic SAW structures for two different piezoelectric substrates. Then we compare measured and simulated input admittances of structures similar to SAW resonators.

FEM-Based determination of real and complex elastic, dielectric, and piezoelectric moduli in piezoceramic materials

We propose an enhanced iterative scheme for the precise reconstruction of piezoelectric material parameters from electric impedance and mechanical displacement measurements. It is based on finite-element simulations of the full three-dimensional piezoelectric equations, combined with an inexact Newton or nonlinear Landweber iterative inversion scheme. We apply our method to two piezoelectric materials and test its performance. For the first material, the manufacturer provides a full data set; for the second one, no material data set is available. For both cases, our inverse scheme, using electric impedance measurements as input data, performs well.

Efficient Modeling of Ferroelectric Behavior for the Analysis of Piezoceramic Actuators

This work proposes a method of efficiently modeling the hysteresis of ferroelectric materials. Our approach includes the additive combination of a reversible and an irreversible portion of the polarization and strain, respectively. Whereas the reversible parts correspond to the common piezoelectric linear equations, the irreversible parts are modeled by hysteresis operators. These operators are based on Preisach and Jiles-Atherton hysteresis models which are well-established tools in ferromagnetic modeling. In contrast to micromechanical approaches, a Preisach or a Jiles-Atherton hysteresis operator can be efficiently numerically evaluated. A comparison of the resulting simulations to measured data concludes the article.

Finite element simulation of nonlinear wave propagation in thermoviscous fluids including dissipation

A recently developed finite element method (FEM) for the numerical simulation of nonlinear sound wave propagation in thermoviscous fluids is presented. Based on the nonlinear wave equation as derived by Kuznetsov, typical effects associated with nonlinear acoustics, such as generation of higher harmonics and dissipation resulting from the propagation of a finite amplitude wave through a thermoviscous medium, are covered. An efficient time-stepping algorithm based on a modification of the standard Newmark method is used for solving the nonlinear semidiscrete equation system. The method is verified by comparison with the well-known Fubini and Fay solutions for plane wave problems, where good agreement is found. As a practical application, a high intensity focused ultrasound (HIFU) source is considered. Impedance simulations of the piezoelectric transducer and the complete HIFU source loaded with air and water are performed and compared with measured data. Measurements of radiated low and high amplitude pressure pulses are compared with corresponding simulation results. The obtained good agreement demonstrates validity and applicability of the nonlinear FEM.

An efficient calculation scheme for the numerical simulation of coupled magnetomechanical systems

A recently developed modeling scheme for the numerical simulation of coupled magnetomechanical systems immersed in an acoustic fluid is presented. The scheme allows the calculation of dynamic rigid motions as well as deformations of magnetic and anti-magnetic materials in a magnetic field. The equations governing the magnetic, mechanical and acoustic field quantities are solved using a combined finite-element-boundary-element-method (FEM-BEM), resulting in a separation of the stationary and the moving parts of the structure. Therewith, the well known problem of mesh distortion in finite element techniques due to moving parts can be avoided. A computer simulation of a magnetomechanical transducer immersed in an acoustic fluid (acoustic power source) is presented demonstrating the efficiency of the developed algorithm.

Analytical analysis and finite element simulation of advanced membranes for silicon microphones

In this paper, advanced membrane designs are simulated in order to improve the sensitivity of micromachined silicon condenser microphones. Analytical analyzes and finite element simulations have been carried out to derive algebraic expressions for the mechanical compliance of corrugated membranes and membranes supported at spring elements. It is shown that the compliance of both types of membranes can be modeled with the help of an enhanced theory of circular membranes. For spring membranes, a numerically derived and design dependent constant takes into account the reduced suspension. The mechanical stress in corrugated membranes is calculated using a geometrical model and is confirmed by finite element simulations. A very good agreement between theory and experimental results is demonstrated for spring membranes of different shape and for membranes with varying number of corrugations. In a silicon microphone application, a high electro-acoustical sensitivity up to 8.2 mV/Pa/V is achieved with a membrane diameter of only 1 mm.

Finite element analysis of coupled electromagnetic acoustic systems

Electromagnetic acoustic transducers (EMATs) are used for nondestructive testing of electric conductive materials. In this paper, the authors introduce an efficient numerical calculation scheme, based on the finite element method, which is capable of simulating both the transmission and detection of ultrasonic waves by EMATs. Therefore, the generation of waves by electromagnetic forces (transmitting case), as well as the induced voltage resulting from the magnetic field of eddy currents, which are generated by the waves in a constant magnetic field, can be calculated. 2D computer simulations and measured data obtained with a pair of matched plate wave EMATs are presented to show the applicability of the developed calculation scheme.

Finite element modeling of the pulse-echo behavior of ultrasound transducers

A new method for the numerical computation of the pulse-echo behavior of ultrasound transducers is presented. In this numerical scheme, the given physical problem is split into two finite element models for which three subsequential computer runs are performed. In a first finite element model, the transducer with its fluid environment is set up and, in a second finite element mesh an obstacle reflecting the outgoing ultrasound wave is modelled. The wave propagation in the fluid area between the first and the second mesh is computed via Helmholtz integral. The reported numerical scheme is applicable to any type of piezoelectric transducer. In this paper, the pulse-echo voltages due to reflecting elastic obstacles with different geometries are presented for ultrasound antennas which are of practical interest in respect to medical imaging. The obtained simulation results are in good agreement with experimental data. The new method allows the precise computer simulation of complex piezoelectric transducers on a highend PC